Boundedness and Stability of Leslie–Gower Model with Sokol–Howell Functional Response

Ali, Safaa Jawad; Arifin, Norihan Md; Naji, Raid Kamel; Ismail, Fudziah; Bachok, Norfifah

doi:10.1007/978-981-10-0519-0_2

Cited by 5 publications

(7 citation statements)

References 19 publications

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“…study the complex dynamics of a three-species food chain using a Holling Type IV functional response and report chaotic behaviour in a narrow region of the bifurcation parameter space for biologically realistic parameter values. Ali et al (2016b) infer that increasing or decreasing the top predator's intrinsic growth rate can change the dynamics of food chains from chaotic to asymptotically stable. In addition, an increase in the death rate of the middle predator can stabilize the dynamics of this system.…”

Section: One-prey -Two-predator Systemmentioning

confidence: 99%

Ecological perspective of the diversity of functional responses

Pervez

Singh²,

Bozdoğan

2018

European Journal of Environmental Sciences

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Prey-predator interactions have been modelled by numerous workers. Ecologists have continuously modified Lotka-Volterra equations in order to provide more realistic descriptions of the complexity of these interactions. The response of predator(s) to increasing prey density can be best described in terms of a functional response, which is an important criterion determining the success or failure of predator(s) to control fluctuating prey populations. The functional response of a predator is further differentiated into Holling's Type I, II, III, IV and V. We discuss one-prey and one-predator interactions, in which the models are modified by the inclusion of steady-state satiation and growth factors. We review situations where two prey and one predator interact, and vice versa. We also discuss Holling's Type IV model relevant to competition and food chains. There is a need to examine functional responses as these models were mostly developed by pure mathematicians and their relevance to field conditions remains largely untested. Prey-predator interactions can be affected even by small factors and ecologists should include these models in their experimental design when attempting to predict realistic interactions.

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Section: One-prey -Two-predator Systemmentioning

confidence: 99%

Ecological perspective of the diversity of functional responses

Pervez

Singh²,

Bozdoğan

2018

European Journal of Environmental Sciences

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“…The model proposed in takes the following form:

\frac{d x}{d t} = a_{0} x - b_{0} x^{2} - \frac{v_{0} x y}{d_{1} + x^{2}} = x f_{1} (x, y, z),

\frac{d y}{d t} = - a_{1} y + \frac{v_{1} x y}{d_{1} + x^{2}} - \frac{v_{2} y z}{d_{2} + y} = y f_{2} (x, y, z),

\frac{d z}{d t} = c_{3} z^{2} - \frac{v_{3} z^{2}}{d_{3} + y} = z^{2} f_{3} (x, y, z),

with initial conditions

x (0) > 0, y (0) > 0, z (0) > 0 .

The parameter definitions are given in Table .…”

Section: The Proposed Model Systemmentioning

confidence: 99%

“…Recently, Ali et al . have proposed a three‐species food chain model, where a top predator modeled via a modified Leslie–Gower scheme, depredates on a middle predator that in turn depredates on a prey species. The authors model the interaction between middle predator and prey via a simplified Holling type IV (non‐monotonic function response).…”

Section: Introductionmentioning

confidence: 99%

“…The model considered by Ali et al . falls within the general class of models of the following type:

\frac{d x}{d t} = a_{0} x - b_{0} x^{2} - y f_{1} (x, y),

\frac{d y}{d t} = - a_{1} y + ε y f_{1} (x, y) - y f_{2} (y, z),

\frac{d z}{d t} = c_{3} z^{2} - \frac{v_{3} z^{2}}{d_{3} + y}

Upadhyay et al . and Aziz‐Alaoui initiated the study of this class of models when f 1 , f 2 are Holling type II functional response.…”

Section: Introductionmentioning

confidence: 99%

“…Our goal is twofold. We first aim to show that the global boundedness results of are incorrect and then to establish certain results on global existence as well as finite‐time blowup for the model introduced in . We next aim to continue to study the dynamics of the model, in light of Hopf bifurcation, time delay, Turing instability. …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the explosive instability in a three‐species food chain model with modified Holling type IV functional response

Parshad

Upadhyay

Mishra

et al. 2017

Math Methods in App Sciences

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In earlier literature, a version of a classical three‐species food chain model, with modified Holling type IV functional response, is proposed. Results on the global boundedness of solutions to the model system under certain parametric restrictions are derived, and chaotic dynamics is shown. We prove that in fact the model possesses explosive instability, and solutions can explode/blow up in finite time, for certain initial conditions, even under the parametric restrictions of the literature. Furthermore, we derive the Hopf bifurcation criterion, route to chaos, and Turing bifurcation in case of the spatially explicit model. Lastly, we propose, analyze, and simulate a version of the model, incorporating gestation effect, via an appropriate time delay. The delayed model is shown to possess globally bounded solutions, for any initial condition. Copyright © 2017 John Wiley & Sons, Ltd.

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